1. TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLESBy
Arijit Gupta

2. ACUTE ANGLE
Acute Angle:
An angle whose measure is greater than zero but less than 90 is called an “acute angle” T E R M I N A L Y o
Initial ray

3. A c b Ѳ B C a Some Old Houses Can’t Always Hide Their Old Age
Commonly used mnemonic for these ratios :
Some Old Houses Can’t Always Hide
Their Old Age

4. History:
Trigonometric functions(also called circular functions) are functions of an angle.
They are used to relate the angles of a triangle to the lengths of the sides of a triangle.
Sumerian astronomers introduced angle measure, using a division of circles into 360 degrees.
The sine function was first defined in the “surya siddhanta” and its properties were further documented by the fifth century Indian mathematician and astronomer “Aryabhatta”.
By 10th century the six trigonometric functions were used.

5. It is also the foundation of the practical art of surveying
Applications:
In 240 B.C. a mathematician named “Eratosthenes” discovered the radius of the earth as miles using trigonometric functions..
In 2001 a group of European astronomers did an experiment by using trigonometric functions and they got all the measurement, they calculate the Venus was about 105,000,000 km away from the sun and the earth was about 150, 000, 000 km away.
Optics and statics are 2 early fields of Physics that use trigonometry.
It is also the foundation of the practical art of surveying

6. 1. Prove that
Sol:
2. Prove that
Sol:

7. Fundamental Relations:c b Ѳ C B a
Squaring and adding both the equations
From the above diagram, By Pythagorean Rule,

8. A c b Ѳ C B a Squaring and subtracting the equations, we get
From the above diagram, By Pythagorean Rule,
Similarly,

9. Example: Prove that
sol:

10. Ex: Prove that sec2Ѳ - cosec2Ѳ = tan2Ѳ - cot2Ѳ
Sol: We know that sec2Ѳ - tan2Ѳ = 1 = cosec2Ѳ - cot2Ѳ
sec2Ѳ - tan2Ѳ = cosec2Ѳ - cot2Ѳ
sec2Ѳ - cosec2Ѳ = tan2Ѳ - cot2Ѳ
Example: Prove that
Sol: Given that

11. Values of the trigonometrical ratios :00
300
450
600
900
Sin 1
Cos
Tan ∞
Cosec 2
Sec
Cot

12. 5 4 Ѳ 3 Example: find the value of tan450.sec300 - cot900.cosec450
Sol: Given that tan450.sec cot900.cosec450
= =
Example: If cosѲ = 3/5, find the value of the other ratios
Sol: Given that cosѲ = 3/5 = adj / hyp
thus using reference triangle adj = 3, hyp = 5,by Pythagorean principle opp = 4 5 4 Ѳ 3

13. THANK YOU HAPPY CALCULATING